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A constructive theory of the numerically accessible manybody localized to thermal crossover
by Philip J D Crowley, Anushya Chandran
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Philip Crowley 
Submission information  

Preprint Link:  scipost_202107_00043v2 (pdf) 
Date accepted:  20220610 
Date submitted:  20220511 16:03 
Submitted by:  Crowley, Philip 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
The manybody localised (MBL) to thermal crossover observed in exact diagonalisation studies remains poorly understood as the accessible system sizes are too small to be in an asymptotic scaling regime. We develop a model of the crossover in short 1D chains in which the MBL phase is destabilised by the formation of manybody resonances. The model reproduces several properties of the numerically observed crossover, including an apparent correlation length exponent $\nu=1$, exponential growth of the Thouless time with disorder strength, linear drift of the critical disorder strength with system size, scalefree resonances, apparent $1/\omega$ dependence of disorderaveraged spectral functions, and subthermal entanglement entropy of small subsystems. In the crossover, resonances induced by a local perturbation are rare at numerically accessible system sizes $L$ which are smaller than a \emph{resonance length} $\lambda$. For $L \gg \sqrt{\lambda}$ (in lattice units), resonances typically overlap, and this model does not describe the asymptotic transition. The model further reproduces controversial numerical observations which Refs. [Suntajs et al 2019, Sels & Polkovnikov 2020] claimed to be inconsistent with MBL. We thus argue that the numerics to date is consistent with a MBL phase in the thermodynamic limit.
Author comments upon resubmission
We thank both referees for their time and careful reading of our manuscript. Report 2 recommends publication as is. We detail the changes we have made to address the comments raised in report 1 below.
We hope with this response, we have satisfied the queries of the referees and editors.
 “To my taste the results of Refs. [1,2] are sometimes referred to in a bit subjective way, which doesn't always appear appropriate. For instance, on page 2 it is written that both Refs. [1,2] "claim that the numerical data precludes the possibility of an MBL phase altogether". I don't think that both papers actually claim that, the claims are partly weaker. Maybe it would be possible to weaken such statements slightly. By the way, both are now also published in journals.”
We have substituted the line “claim that the numerical data precludes the possibility of an MBL phase altogether” to “argue that finite size numerics is inconsistent with the existence of MBL in the thermodynamic limit”, as paraphrased from the conclusion of Ref [2].
 “When I understand correctly (see for instance Eq. (43)), the local perturbation studied in this manuscript is always in a weak coupling regime. Is that correct? In any case it would be good to clarify that prominently.”
If by “weak coupling” the referee means \Omega, W >> J, yes, we are always working in this regime. This is where both the MBL finite size crossover and, at even stronger disorder, it is believed the MBL transition also occurs (they certainly do not occur outside of this regime). This regime is introduced in the first sentence “Interacting onedimensional quantum systems generically manybody localise (MBL) in the presence of strong disorder.”. It is further restated below Eq.4 in which the model is introduced “We assume two key properties of H(t): (i) it has no global conservation laws, and (ii) for some finite \Omega, W >> J, the model is Floquet manybody localised, as per Ref. [71]. The specific form of H(t) is otherwise unimportant.”
If the referee instead means perturbatively weak coupling, then no, we do not make this assumption. Our results are nonperturbative in the probe spin coupling, this is important in allowing us to capture the effect of the nonperturbatively corrected states or “resonances”. We note this picture of resonance formation we assume is validated in a more controlled model in Ref [72].
 “Section 2.3 on the thermal phase is rather brief. Although (as the authors say) the RM is not applicable, in the next sentence they say that RM still holds. This sounds a bit confusing and it would be good to clarify this. Further, the authors mention "almostlbits" without any reference. I feel that it is not clear what kind of lbits these should be.”
Precisely which of the predictions of the RM we expect to hold, and for what regime of time is clarified by the sentence “Despite being generally inapplicable, the early time predictions of the RM are found to hold even in the thermal regime” and following equations. The almost lbits are defined in the manuscript Eq. 60 and surrounding text. Where we write “almostlbits [are] operators [which] have the same properties as lbits (mutually commuting exponentially localised etc.), but only “almost commute” with the Hamiltonian: [H,\tau]<\omega_\xi.
We are not the first to propose such objects. We have included a citation to the earlier work Ref. [73] in which similar objects were proposed.
List of changes
 Substituted the line “claim that the numerical data precludes the possibility of an MBL phase altogether” to “argue that finite size numerics is inconsistent with the existence of MBL in the thermodynamic limit”.
 Included citation to Ref. [73]
Published as SciPost Phys. 12, 201 (2022)